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March 16th, 2010, 04:58 AM   #1
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Twin primes

If n and n+2 are twin primes, there are other two couples of twin primes (l, l+2, m and m+2) so that:

n = l+m+1

Is it through?
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March 16th, 2010, 06:08 AM   #2
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Re: Twin primes

Quote:
Originally Posted by loes
If n and n+2 are twin primes, there are other two couples of twin primes (l, l+2, m and m+2) so that:

n = l+m+1

Is it through?
First, I trust you mean "prime", not "twin prime"; the only pair (n, n+2) of twin primes is (3, 5). A twin prime is a prime p for which p+2 is prime.

First cut: "If n is a twin prime, there exist twin primes l and m with n = l + m + 1." This is false; take n = 3.

Second cut: "Do there exist twin primes l, m, n with n = l + m + 1?" This is true; take (l, m, n) = (5, 5, 11).

Third cut: "There exist infinitely many twin primes n such that n = l + m + 1, with l and m twin primes." This cannot be proven with present techniques; it is stronger than the twin prime conjecture.

Can you clarify what you meant, if it wasn't one of these guesses?
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March 16th, 2010, 11:43 PM   #3
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Re: Twin primes

Please, let consider the question in a different way.

1 3 5 11 17 29 41 59 227 ...
2 4 6 12 18 30 42 60 228 ...
3 5 7 13 19 31 43 61 229 ...

Is it through that each number in the central line can be written as a sum of other two numbers of the same line?
e.g. 30 = 18 + 12 so that 29 = 17 + 11 + 1 and 31 = 19 + 13 1
228 = 198 + 30 so that 227 = 197 + 29 + 1 and 229 = 199 + 31 - 1
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March 17th, 2010, 04:46 AM   #4
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Re: Twin primes

Are even numbers the sum of two odd numbers? Yes. But I don't know if that's what you intended.
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March 17th, 2010, 05:37 AM   #5
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Re: Twin primes

Each even number is the average of twin prime (e.g. 12 = (11+13)/2)

Then, 1,3 -> 2, 3,5 -> 4, 5,7->6....,17,19->18,...,59,61->60,....

We have then a sequence,

2, 4, 6, 12, 18, 30,...,60,...

My question is if is it through that each number of the sequence can be written as the sum of other two numbers of the same sequence.
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March 17th, 2010, 06:41 AM   #6
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Re: Twin primes

Quote:
Originally Posted by loes
Each even number is the average of twin prime (e.g. 12 = (11+13)/2)
I'll assume you mean "the second list is averages of prime twins", since the literal statement you give is false. Terminology note: the twin primes are 3, 5, 11, 17, 29, ... while the prime twins are (3, 5), (5, 7), (11, 13), (17, 19), ....

Quote:
Originally Posted by loes
Then, 1,3 -> 2, 3,5 -> 4, 5,7->6....,17,19->18,...,59,61->60,....
I see you're counting 1 as a prime. That was typical 100 years ago but it's rare now. (In terms of abstract algebra, 1 is a unit of Z, not a prime element.)

Quote:
Originally Posted by loes
2, 4, 6, 12, 18, 30,...,60,...
Sloane's A167777: 2 along with A014574.

Quote:
Originally Posted by loes
My question is if is it through that each number of the sequence can be written as the sum of other two numbers of the same sequence.
It seems that (other than 2) all elements of A167777 can be written as the sum of two members of A167777.
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March 25th, 2010, 01:19 AM   #7
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Re: Twin primes

If all elements of A167777 can be written as the sum of two members of A167777,it means that each twin prime is a function of two other twin primes.
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March 25th, 2010, 10:18 AM   #8
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Re: Twin primes

Quote:
Originally Posted by loes
each twin prime is a function of two other twin primes.
This is unconditionally true. For example, take f(m, n) = {
if m = 3, the first twin prime greater than n;
if m = 5, the first twin prime less than n, or 0 if none exists;
otherwise, 0.
}

For every twin prime p (here meaning members of A001097), there exist two different twin primes q and r such that p = f(q, r):

3 = f(5, 5)
5 = f(3, 3)
7 = f(3, 5)
11 = f(3, 7)
. . .
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